Beyond IGO-Flow: Toward Convergence Analysis of IGO in Continuous Spaces

2026-06-16

AI Summary

This work addresses the lack of convergence theory for discrete-time Information-Geometric Optimization (IGO) in continuous spaces under practical settings involving non-infinitesimal learning rates, full covariance adaptation, and quantile-based reweighting. Focusing on natural gradient updates over the family of multivariate Gaussian distributions optimizing strongly convex quadratic objectives, the paper establishes the first convergence guarantees for both the mean vector and the covariance matrix under a unified setting that simultaneously incorporates a fixed positive learning rate, full covariance adaptation, and quantile weighting. By leveraging tools from information geometry, expectation parameters, and matrix condition number analysis, the authors prove that the covariance matrix converges to the zero matrix, and that the mean converges to the global optimum whenever the condition number of the scaled covariance remains frequently bounded—thereby bridging a critical theoretical gap between IGO frameworks and practical algorithms such as CMA-ES.

Abstract

Information-Geometric Optimization (IGO) provides a unified framework for black-box optimization by interpreting the adaptation of a search distribution as a natural gradient update. Despite its conceptual importance, the convergence theory of IGO remains limited: most existing results concern continuous-time idealizations such as the IGO flow, rather than discrete-time updates with non-infinitesimal learning rates. In this paper, we study discrete-time IGO in continuous spaces, formulated as natural gradient updates in the expectation-parameter coordinates of an exponential family. In particular, we analyze IGO over the multivariate Gaussian family on strongly convex quadratic objective functions. Our analysis covers a setting that simultaneously incorporates full covariance adaptation, a fixed positive learning rate, and quantile-based weights. In this setting, we prove that the covariance matrix converges to the zero matrix. We further show that the mean vector converges to the global optimum, provided that the condition number of the appropriately scaled covariance matrix is bounded at sufficiently frequent iterations. These results advance the convergence theory of IGO and help bridge the gap between the mathematical theory of IGO and practical covariance-adaptive search methods such as CMA-ES.

Problem

• Information-Geometric Optimization
• convergence analysis
• discrete-time updates
• natural gradient
• covariance adaptation

Innovation

• Information-Geometric Optimization
• natural gradient
• covariance adaptation
• convergence analysis
• discrete-time IGO